L(s) = 1 | − 1.81·2-s − 0.874·3-s + 1.27·4-s − 5-s + 1.58·6-s + 7-s + 1.30·8-s − 2.23·9-s + 1.81·10-s − 4.01·11-s − 1.11·12-s + 3.22·13-s − 1.81·14-s + 0.874·15-s − 4.92·16-s + 1.62·17-s + 4.04·18-s − 4.71·19-s − 1.27·20-s − 0.874·21-s + 7.27·22-s + 0.483·23-s − 1.14·24-s + 25-s − 5.83·26-s + 4.57·27-s + 1.27·28-s + ⋯ |
L(s) = 1 | − 1.27·2-s − 0.504·3-s + 0.638·4-s − 0.447·5-s + 0.646·6-s + 0.377·7-s + 0.463·8-s − 0.745·9-s + 0.572·10-s − 1.21·11-s − 0.322·12-s + 0.893·13-s − 0.483·14-s + 0.225·15-s − 1.23·16-s + 0.394·17-s + 0.953·18-s − 1.08·19-s − 0.285·20-s − 0.190·21-s + 1.55·22-s + 0.100·23-s − 0.233·24-s + 0.200·25-s − 1.14·26-s + 0.881·27-s + 0.241·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 3 | \( 1 + 0.874T + 3T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 - 0.483T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 - 8.72T + 37T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 4.28T + 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 - 0.252T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 5.95T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 + 9.36T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57375819921264103694187908283, −7.20546653490586299021698925468, −6.00877536275057668892904466250, −5.65279797288710084896501458125, −4.67520406738963294182070701876, −3.98793270751809707814952465326, −2.84966242374402092480384585526, −1.99153963251898467452226893519, −0.859523045529503653893959527249, 0,
0.859523045529503653893959527249, 1.99153963251898467452226893519, 2.84966242374402092480384585526, 3.98793270751809707814952465326, 4.67520406738963294182070701876, 5.65279797288710084896501458125, 6.00877536275057668892904466250, 7.20546653490586299021698925468, 7.57375819921264103694187908283